Find the largest domain in C where the function fz Log zz1

Find the largest domain in C where the function f(z) = Log (z/z+1) is analytic. Justify your answer. Here Log z = ln |z| + iArg z denotes the principal value of the complex logarithm

Solution

note : log z = ln modz+ i Argz

for the function to be analytic f\'z must exist

f\'z = z+1-z/(z+1)^2 +i0

f\'z = 1/z(z+1)

for f\'z to exist z(z+1) not equal to 0

hence Z belongs to C -{0,-1}